arXiv subjects

Xuenan Li

Publications and source records attributed to Xuenan Li.

9 recordsLinked to original sources

arXiv2026

Pseudo-magnetism in a strained discrete honeycomb lattice

Slowly varying nonuniform strains of non-magnetic wave propagating media with honeycomb symmetry induce an effective- or pseudo-magnetic field, a phenomenon observed first in graphene, and later in photonic crystals and other physical settings. Starting with a discrete nearest-neighbor tight-binding model of a non-uniformly strained honeycomb medium, we derive the continuum effective magnetic Dirac Hamiltonian governing the envelope dynamics of wave packets, which are spectrally localized near a Dirac point (conical band degeneracy) of the unperturbed honeycomb. For unidirectional deformations of bounded gradient, which preserve translation invariance along the ''armchair'' direction, we prove the existence of time-harmonic states which are plane-wave like (pseudo-periodic) along the armchair direction and exponentially localized transverse to it. We also obtain the leading order multi-scale structure of such modes for small deformation gradients. Their transverse localization are determined by the eigenstates of a one dimensional effective Dirac Hamiltonian. Our rigorous results apply to deformations which induce an approximate perpendicular constant pseudo-magnetic field (Landau gauge), and yield states with nearly flat band (Landau level) spectrum and hence very high density of states. In contrast, the analogous deformation which preserves translations in the zigzag direction induces no such localization. Corroborating numerical simulations for the different deformation types are presented.

math-ph↗

arXiv2025

An Asymptotic Approach for Modeling Multiscale Complex Fluids at the Fast Relaxation Limit

We present a new asymptotic strategy for general micro-macro models which analyze complex viscoelastic fluids governed by coupled multiscale dynamics. In such models, the elastic stress appearing in the macroscopic continuum equation is derived from the microscopic kinetic theory, which makes direct numerical simulations computationally expensive. To address this challenge, we introduce a formal asymptotic scheme that expands the density function around an equilibrium distribution, thereby reducing the high computational cost associated with the fully coupled microscopic processes while still maintaining the dynamic microscopic feedback in explicit expressions. The proposed asymptotic expansion is based on a detailed physical scaling law which characterizes the multiscale balance at the fast relaxation limit of the microscopic state. An asymptotic closure model for the macroscopic fluid equation is then derived according to the explicit asymptotic density expansion. Furthermore, the resulting closure model preserves the energy-dissipation law inherited from the original fully coupled multiscale system. Numerical experiments are performed to validate the asymptotic density formula and the corresponding flow velocity equations in several micro-macro models. This new asymptotic strategy offers a promising approach for efficient computations of a wide range of multiscale complex fluids.

math-ph↗

arXiv2025

A nonlinear homogenization-based perspective on the soft modes and effective energies of some conformal metamaterials

There is a growing mechanics literature concerning the macroscopic properties of mechanism-based mechanical metamaterials. This amounts mathematically to a homogenization problem involving nonlinear elasticity. A key goal is to identify the "soft modes" of the metamaterial. We achieve this goal using methods from homogenization for some specific 2D examples -- including discrete models of the Rotating Squares metamaterial and the Kagome metamaterial -- whose soft modes are compressive conformal maps. The innovation behind this achievement is a new technique for bounding the effective energy from below, which takes advantage of the metamaterial's structure and symmetry.

math.AP↗

arXiv2025

A Constrained Saddle Search Approach for Constructing Singular and Flexible Bar Frameworks

Singularity analysis is essential in robot kinematics, as singular configurations cause loss of control and kinematic indeterminacy. This paper models singularities in bar frameworks as saddle points on constrained manifolds. Given an under-constrained, non-singular bar framework, by allowing one edge to vary its length while fixing lengths of others, we define the squared length of the free edge as an energy functional and show that its local saddle points correspond to singular and flexible frameworks. Using our constrained saddle search approach, we identify previously unknown singular and flexible bar frameworks, providing new insights into singular robotics design and analysis.

cs.RO↗

arXiv2025

The effective energy of a lattice metamaterial

We study the sense in which the continuum limit of a broad class of discrete materials with periodic structures can be viewed as a nonlinear elastic material. While we are not the first to consider this question, our treatment is more general and more physical than those in the literature. Indeed, it applies to a broad class of systems, including ones that possess mechanisms; and we discuss how the degeneracy that plagues prior work in this area can be avoided by penalizing change of orientation. A key motivation for this work is its relevance to mechanism-based mechanical metamaterials. Such systems often have ``soft modes'', achieved in typical examples by modulating mechanisms. Our results permit the following more general definition of a soft mode: it is a macroscopic deformation whose effective energy vanishes -- in other words, one whose spatially-averaged elastic energy tends to zero in the continuum limit.

math-ph↗

arXiv2025

A Constrained Optimization Approach for Constructing Rigid Bar Frameworks with Higher-order Rigidity

We present a systematic approach for constructing bar frameworks that are rigid but not first-order rigid, using constrained optimization. We show that prestress stable (but not first-order rigid) frameworks arise as the solution to a simple optimization problem, which asks to maximize or minimize the length of one edge while keep the other edge lengths fixed. By starting with a random first-order rigid framework, we can thus design a wide variety of prestress stable frameworks, which, unlike many examples known in the literature, have no special symmetries. We then show how to incorporate a bifurcation method to design frameworks that are third-order rigid. Our results highlight connections between concepts in rigidity theory and constrained optimization, offering new insights into the construction and analysis of bar frameworks with higher-order rigidity.

math.MG↗

arXiv2023

Some results on the Guest-Hutchinson modes and periodic mechanisms of the Kagome lattice metamaterial

Lattice materials are interesting mechanical metamaterials, and their mechanical properties are often related to the presence of mechanisms. The existence of periodic mechanisms can be indicated by the presence of Guest-Hutchinson (GH) modes, since GH modes are sometimes infinitesimal versions of periodic mechanisms. However, not every GH mode comes from a periodic mechanism. This paper focuses on: (1) clarifying the relationship between GH modes and periodic mechanisms; and (2) answering the question: which GH modes come from periodic mechanisms? We focus primarily on a special lattice system, the Kagome lattice. Our results include explicit formulas for all two-periodic mechanisms of the Kagome lattice, and a necessary condition for a GH mode to come from a periodic mechanism in general. We apply our necessary condition to the two-periodic GH modes, and also to some special GH modes found by Fleck and Hutchinson using Bloch-type boundary conditions on the unit cell of the Kagome lattice.

math-ph↗

arXiv2001

Parallel dynamics and computational complexity of the Bak-Sneppen model

The parallel computational complexity of the Bak-Sneppen evolution model is studied. It is shown that Bak-Sneppen histories can be generated by a massively parallel computer in a time that is polylogarithmic in the length of the history. In this parallel dynamics, histories are built up via a nested hierarchy of avalanches. Stated in another way, the main result is that the logical depth of producing a Bak-Sneppen history is exponentially less than the length of the history. This finding is surprising because the self-organized critical state of the Bak-Sneppen model has long range correlations in time and space that appear to imply that the dynamics is sequential and history dependent. The parallel dynamics for generating Bak-Sneppen histories is contrasted to standard Bak-Sneppen dynamics. Standard dynamics and an alternate method for generating histories, conditional dynamics, are both shown to be related to P-complete natural decision problems implying that they cannot be efficiently implemented in parallel.

cond-mat.stat-mech↗

arXiv2000

Critical dynamics of two-replica cluster algorithms

The dynamic critical behavior of the two-replica cluster algorithm is studied. Several versions of the algorithm are applied to the two-dimensional, square lattice Ising model with a staggered field. The dynamic exponent for the full algorithm is found to be less than 0.4. It is found that odd translations of one replica with respect to the other together with global flips are essential for obtaining a small value of the dynamic exponent.

physics.comp-ph↗